This simulation shows the behavior of the adaptive hybrid attitude-feedback controller (REF ) for different values of the gap exceeding and how this behaviour is affected by noisy quaternion measurements and time-varying torque disturbances. The desired trajectory is generated by (REF ) with the initial condition \(q_d^T(0)=[1,0_{1\times 3}]\)  and the desired angular velocity \(\omega _d = 0.1\sin (0.2\pi t) \left[ 1,1,1 \right]^{T}\)  \(\left[ \mathrm {rad}/s \right]\)  used in [1]}.
The initial conditions were \(q(0)=[0,0,1,0]^T\) , \(\omega (0)=\bar{u}\)  with \(\bar{u}=u/\Vert u\Vert \)  and \(u=[1, 2, 3]^T\) . The initial discrete state was set \(h(0)=1\)  for all scenarios. The controller gains (REF ) were \(k_v = 3\)  and, \(k_p=0.7\) , with a filter gain (REF ), \(K_f=0.1I_4\) , and the adaptation gain (REF ) was \(\Gamma = \mathrm {diag}\lbrace 1000I_3, I_6\rbrace \) . The unknown inertial matrix (REF ) and the noisy quaternion measurement were the same as in the first simulation. In addition, a torque disturbance \(\dot{p}=v\)  was added (see (REF )), with the initial condition \(p(0)=[0.2,-0.1,-0.05]^{T} \ [\mathrm {Nm}]\) , where \(v=0_{3\times 1}\)  for a constant disturbance and \(v \in \mathbb {R}^{4}\)  whose element is a zero-mean Gauss distribution with variance \(0.2\)  for a time-varying disturbance (see Table REF ).
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